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Theorem dfrnf 4689
Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfrnf.1 𝑥𝐴
dfrnf.2 𝑦𝐴
Assertion
Ref Expression
dfrnf ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrn2 4637 . 2 ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2 nfcv 2229 . . . . 5 𝑥𝑣
3 dfrnf.1 . . . . 5 𝑥𝐴
4 nfcv 2229 . . . . 5 𝑥𝑤
52, 3, 4nfbr 3895 . . . 4 𝑥 𝑣𝐴𝑤
6 nfv 1467 . . . 4 𝑣 𝑥𝐴𝑤
7 breq1 3854 . . . 4 (𝑣 = 𝑥 → (𝑣𝐴𝑤𝑥𝐴𝑤))
85, 6, 7cbvex 1687 . . 3 (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
98abbii 2204 . 2 {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10 nfcv 2229 . . . . 5 𝑦𝑥
11 dfrnf.2 . . . . 5 𝑦𝐴
12 nfcv 2229 . . . . 5 𝑦𝑤
1310, 11, 12nfbr 3895 . . . 4 𝑦 𝑥𝐴𝑤
1413nfex 1574 . . 3 𝑦𝑥 𝑥𝐴𝑤
15 nfv 1467 . . 3 𝑤𝑥 𝑥𝐴𝑦
16 breq2 3855 . . . 4 (𝑤 = 𝑦 → (𝑥𝐴𝑤𝑥𝐴𝑦))
1716exbidv 1754 . . 3 (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
1814, 15, 17cbvab 2211 . 2 {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
191, 9, 183eqtri 2113 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1290  wex 1427  {cab 2075  wnfc 2216   class class class wbr 3851  ran crn 4453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-cnv 4460  df-dm 4462  df-rn 4463
This theorem is referenced by:  rnopab  4695
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